ARS needs to include both the design and random variables, the number of
variables is relatively large, often exceeding 15 variables. This excludes the use of many
DOE, such as Central Composite Design (CCD). The CCD has 2n vertices, 2n axial
points, and one center point, so the required number of design points is 2n+2n+1,where n
is the number of variables involved. A polynomial of nmth order in terms of n variables has
L coefficients, where
(n + 1)(n + 2)...(n + m)
L = (3-7)
For n = 15, CCD requires 32799 analyses. On the other hand, a quadratic polynomial in
15-variable has 136 coefficients. From our experience, in order to estimate these
coefficients, the number of analyses only needs to be about twice as large as the number
of coefficients, which is less than one percent of the number of vertices for 15-variable
space. Therefore, other DOEs such as CCD using fractional factorial design (Myers and
Montgomery 1995) need to be used. The fractional factorial CCD is intended for the
construction of quadratic RSA. Orthogonal arrays (Myers and Montgomery 1995) are
used for the construction of higher order RS (Balabanov 1997 and Padmanabhan et al.
2000). Isukapalli (1999) employed orthogonal arrays to construct SRS. For problems
where only very limited number of analyses is computationally affordable, Box-Behnken
designs or saturated designs can be used (Khuri and Cornell 1996).
In the paper of Qu et al. (2000), it is shown that Latin Hypercube sampling is more
efficient and flexible than orthogonal arrays. The idea of Latin Hypercube sampling can
be explained as follows: assume that we want n samples out of k random variables. First,
the range of each random variable is divided into n nonoverlapping intervals on the basis
of equal probability. Then one value is selected randomly from each interval. Finally, by